Given a semi discrete system dU/dt = F(U). where U and F are large vectors. If this system results from method of lines for nonlinear PDEs that involves upwinding, then the actual form of dependence of F(U) on components of U is not known a priori, because they are only computed at run time due to upwinding. In this case, how does one compute derivatives of F(U) with respect to the components of U? I will be more than happy if anyone can suggest a way on how to compute these derivatives. Thanks.
This is programming problem. At each time step one has to see the direction of flow and accordingly derivatives are computed. There will be if .... else statement for up winding.
Yes, programming is not out of it. But let me be specific : In a nonlinear PDE system, F(U) will be the contribution of fluxes and sources, so if an approximate Riemann solver is being used to compute fluxes, then F(U) cannot be know before hand because one does not tell what exactly the Riemann solver will return for each component of F(U).
What I feel is , you have sources from them find fluxes. Use them in F(U) to obtain new values of sources and march ahead. It can not be half here and half there. You need to find all fluxes first to use F(U).
Write your scheme as a central difference scheme plus an artificial diffusion term. The diffusion coefficient will be non-linear but you can differentiate it w.r.t U. This way you avoid if-else statements. Or, depending on what you want to use it for, compute it numerically.
Computing F(U) isn't a problem, my program is able to do it perfectly but the concern is that the program does the computation of F(U) at run time that a user or implementer cannot know what is returned each time. In any case, I will find a way around the problem. The whole idea is to be able to implement implicit time integration. So, I could avoid newton method which needs derivatives. Or I can do more analysis on my Riemann solvers to see if I can formulate a procedure for the needed derivatives. Thanks for your contribution.
I think you have to be clearer. I thought the problem was that you are using upwinding, i.e., a non-linear stencil that changes in space. If so, my previous answer stands. If your aim is to write an ODE solver that works irrespective of F(U) then it is a different matter. Now you are also saying that the aim is implicit time-stepping which I suspected from the beginning. In that case you have two options: 1) You calculate F'(U) analytically and code it. That is doable for simple second-order schemes. 2) Calculate F'(U) numerically. In this case it, at least to some degree, does not matter how F(U) is obtained. I would recommend Newton's method and that you have a look at:
http://mdolab.engin.umich.edu/content/computing-derivatives-using-dual-numbers-fortran
Magnus, I didn't see your previous answer before my last comment. So my comment was directed to Mittal's answers.
I think my question has been clear and haven't changed. And the goal has also been put forward in my last comments. I will add that directly computing F'(U) in this case is impossible, because a lot of algorithms are implement to provide each component of it. F here has thousands of components and each is calculated at run time. So that is the challenge. Anyway thanks for your answers and references. I will get back here after more trials. Thanks.
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